What happens to the above
inequality when I multiply through by –1?
The temptation is to say that the answer will be "–3
> –2". But –3 is not greater than –2;
it is in actuality smaller. That is, the correct inequality is
actually the following:

–3 < –2

As you can see, multiplying
by a negative ("–1",
in this case) flipped the inequality sign from "greater than"
to "less than". This is the new wrinkle for solving inequalities:

When solving inequalities,
if you multiply or divide through by a negative, you must also flip
the inequality sign.

To solve "–2x < 5", I need
to divide through by a negative ("–2"),
so I will need to flip the inequality:

Then the solution is: x > –5/2

Solve(2x – 3)/4<2.

First, I'll multiply
through by 4.
Since the "4"
is positive, I don't have to flip the inequality sign:

(2x – 3)/4< 2
(4)
× (2x – 3)/4< (4)(2)
2x – 3 < 8
2x< 11x<11/2 = 5.5

You can use the Mathway widget below to
practice solving a linear inequality. Try the entered exercise, or
type in your own exercise. Then click "Answer" to compare your
answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "View Steps"
on the widget's answer screen will take you to the Mathway site, where
you can register for a free seven-day trial of the software.)

This is what is called
a "compound inequality". It works just like regular inequalities,
except that it has three "sides". So, for instance, when I
go to subtract the 4,
I will have to subtract it from all three "sides".

10 < 3x + 4 < 19
6 < 3x< 152<x<5

You can use the Mathway widget below to
practice solving an interval inequality. Try the entered exercise, or
type in your own exercise. Then click "Answer" to compare your
answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "View Steps"
on the widget's answer screen will take you to the Mathway site, where
you can register for a free seven-day trial of the software.)